Most so-called "professionals" in our area want to simply lay down another layer of 1/4 inch plywood over the existing floor an call it good. I guess if one has no squeeks and can live with the additional build up of flooring it might be ok. I have also seen new linoleum placed directly over the old with what appeared to be good results. In your case you need to get down to the flooring in order to screw or nail the floor solid again. I think Jay J has the right idea, and that is to remove the 1/4 inch underlayment (if you have one) and start over with new. If you are already down to the subfloor, and need to remove the backing that is still stuck to the adhesive after pulling up the linoleum, the only way I think will work is mechanical methods (scraping, scraping, sanding, etc.) Stripers will not work since they will not soak through the backing material to get at the adhesive. I need to do the same thing to my kitchen in the next few weeks, and I'm going to try something different. I'm going to drill small (1/8 inch holes) through the linoleum to the underlayment and try injecting air at about 100psi to see if it will separate the linoleum from the subfloor. It probably won't work, but I will report my findings. No magic solutions here-sorry. Good Luck!